Given point A (2, 4) on the triangle in Quadrant I, how would you describe the coordinates of the new point when the triangle is rotated 90 degrees clockwise?
A. (-2, -4)
B. (4, -2)
C. (-4, 2)
D. (-4, -2)
To describe the coordinates of the new point when the triangle is rotated 90 degrees clockwise, we can use the concept of a rotation matrix.
The rotation matrix for a 90-degree clockwise rotation is:
| cosθ -sinθ |
| sinθ cosθ |
In this case, since we need a 90-degree clockwise rotation, θ (theta) will be -90 degrees.
Substituting the values into the rotation matrix:
| cos(-90) -sin(-90) |
| sin(-90) cos(-90) |
cos(-90) = 0
sin(-90) = -1
cos(-90) = 0
sin(-90) = -1
Therefore, the rotation matrix becomes:
| 0 -(-1) |
| -1 0 |
Using the rotation matrix, we can apply it to the coordinates of point A (2, 4):
| 0 -(-1) | | 2 | | x' |
| -1 0 | * | 4 | = | y' |
Simplifying the matrix multiplication, we get:
| 0 1 | | 2 | | x' |
| -1 0 | * | 4 | = | y' |
Calculating the matrix multiplication:
0 * 2 + 1 * 4 = x'
-1 * 2 + 0 * 4 = y'
Therefore, the coordinates of the new point (x', y') are:
x' = 0 * 2 + 1 * 4 = 4
y' = -1 * 2 + 0 * 4 = -2
So the answer is option B: (4, -2).